On an spt function for the $4$-th symmetrized crank function

TL;DR

This work defines a smallest part function associated with the $4$-th symmetrized crank, linking it to Garvan's second-order $spt$ via $spt_2(n)=\mu_4(n)-\eta_4(n)$ and derives generating-function relations that connect to $M_4(n)$, $p(n)$, and the symmetrized rank/crank components. It proves two main results using Bailey pair techniques: a $q$-series identity for a double-sum generating function (Theorem 1.1) and explicit formulas for $SPT^{-}(n)$ and $SPT^{+}(n)$ in terms of $M_4(n)$, $p(n)$, $\mu_4(n)$, and $\eta_4(n)$ (Theorem 1.2), including the identity $SPT^{+}(n)-SPT^{-}(n)=\;\text{spt}_2(n)$. The paper also establishes congruences, asymptotics $SPT^{\pm}(n)\sim\frac{\sqrt{3}}{40}n e^{\pi\sqrt{2n/3}}$, and develops finite $spt$ analogs $spt^{\pm}_M(n)$ via a 1-fold Bailey framework, with a combinatorial interpretation and connections to finite $q$-series for the second crank moment.

Abstract

In this paper we find the smallest part function related to the $4$-th symmetrized crank function, corresponding to the one obtained in Patkowski [11] for the $4$-th symmetrized rank function. This provides us with a direct relationship with Garvan's second order smallest part function. We obtain some congruences for these $spt$ functions, as well as asymptotics and inequalities. A finite $q$-series which generates an $spt$ function related to the second crank moment is also stated. This identity has an important relationship to the one obtained by Patkowski [12] related to the second rank moment.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• proof
• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Lemma 2.1
• proof : Proof of Theorem \ref{['thm:ex1']}